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Multivariate risk processes with interacting intensities

Part of: Mathematical economics Special processes

Published online by Cambridge University Press:  01 July 2016

Nicole Bäuerle  and
Rudolf Grübel
Nicole Bäuerle*
Affiliation:
Universität Karlsruhe (TH)
Rudolf Grübel*
Affiliation:
Leibniz Universität Hannover
*
∗ Postal address: Institut für Stochastik, Universität Karlsruhe (TH), D-76128 Karlsruhe, Germany.
∗∗ Postal address: Institut für Mathematische Stochastik, Leibniz Universität Hannover, Postfach 6009, D-30167 Hannover, Germany. Email address: rgrubel@stochastik.uni-hannover.de
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Abstract

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The classical models in risk theory consider a single type of claim. In the insurance business, however, several business lines with separate claim arrival processes appear naturally, and the individual claim processes may not be independent. We introduce a new class of models for such situations, where the underlying counting process is a multivariate continuous-time Markov chain of pure-birth type and the dependency of the components arises from the fact that the birth rate for a specific claim type may depend on the number of claims in the other component processes. Under certain conditions, we obtain a fluid limit, i.e. a functional law of large numbers for these processes. We also investigate the consequences of such results for questions of interest in insurance applications. Several specific subclasses of the general model are discussed in detail and the Cramér asymptotics of the ruin probabilities are derived in particular cases.

Keywords

Cramér asymptoticfluid limitLundberg coefficientmultidimensional birth processprobability of ruinrisk reserve processurn model

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 91B30: Risk theory, insurance
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 578 - 601
Copyright
Copyright © Applied Probability Trust 2008 

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